What Is the Logarithmic Norm?

The logarithmic norm of a matrix A\in\mathbb{C}^{n\times n} (also called the logarithmic derivative) is defined by

\notag       \mu(A) = \displaystyle\lim_{\epsilon \to 0+} \frac{ \|I + \epsilon A\| - 1}{\epsilon},

where the norm is assumed to satisfy \|I\| = 1.

Note that the limit is taken from above. If we take the limit from below then we obtain a generally different quantity: writing \delta = -\epsilon,

\notag   \displaystyle\lim_{\epsilon \to 0-} \frac{ \|I + \epsilon A\| - 1}{\epsilon}    = \lim_{\delta \to 0+} \frac{ \|I - \delta A\| - 1}{-\delta}    = -\mu(-A).

The logarithmic norm is not a matrix norm; indeed it can be negative: \mu(-I) = -1.

The logarithmic norm can also be expressed in terms of the matrix exponential.

Lemma 1. For A\in\mathbb{C}^{n\times n},

\notag \mu(A) = \displaystyle\lim_{\epsilon\to 0+} \frac{\log\, \| \mathrm{e}^{\epsilon A}\|} {\epsilon}        = \lim_{\epsilon\to 0+} \frac{\| \mathrm{e}^{\epsilon A}\| - 1} {\epsilon}.

Properties

We give some basic properties of the logarithmic norm.

It is easy to see that

\notag     -\|A\| \le \mu(A) \le \|A\|. \qquad (1)

For z\in\mathbb{C}, we define \mathrm{sign}(z) = z/|z| for z\ne 0 and \mathrm{sign}(0) = 0.

Lemma 2. For A,B\in\mathbb{C}^{n\times n} and \lambda\in\mathbb{C},

  • \mu(\lambda A) = |\lambda| \mu\bigl(\mathrm{sign}(\lambda)A\bigr),
  • \mu(A + \lambda I) = \mu(A) + \mathrm{Re}\,\lambda,
  • \mu(A + B ) \le \mu(A) + \mu(B),
  • |\mu(A) - \mu(B)| \le \|A - B\|.

The next result shows the crucial property that \mu(A) features in an easily evaluated bound for the norm of \mathrm{e}^{At} and that, moreover, \mu(A) is the smallest constant that can appear in this bound.

Theorem 3. For A\in\mathbb{C}^{n\times n} and a consistent matrix norm,

\notag   \|\mathrm{e}^{At}\| \le \mathrm{e}^{\mu(A)t}, \quad t\ge0.   \qquad (2)

Moreover,

\notag   \mu(A) = \min\{\, \theta\in\mathbb{R}:         \|\mathrm{e}^{At}\| \le \mathrm{e}^{\theta t}               ~\mathrm{for~all}~t\ge 0 \,\}.

Proof. Given any \delta > 0, by Lemma 1 there exists h > 0 such that

\notag     \displaystyle\frac{ \| \mathrm{e}^{At}\| - 1}{t} - \mu(A) < \delta,     \quad t\in[0,h],

or

\notag     \| \mathrm{e}^{At}\| \le 1 + (\mu(A) + \delta)t         \le \mathrm{e}^{(\mu(A) + \delta)t},     \quad t\in[0,h]

(since \mathrm{e}^x \ge 1+x for all x). Then for any integer k, \|\mathrm{e}^{Atk}\| = \|( \mathrm{e}^{At})^k\| \le \| \mathrm{e}^{At}\|^k \le \mathrm{e}^{(\mu(A) + \delta)tk}, and hence \|\mathrm{e}^{At}\| \le \mathrm{e}^{(\mu(A) + \delta)t} holds for all t\in[0,\infty). Since \delta is arbitrary, it follows that \|\mathrm{e}^{At}\| \le \mathrm{e}^{\mu(A)t}.

Finally, if \| \mathrm{e}^{At}\| \le \mathrm{e}^{\theta t} for all t\ge 0 then (\| \mathrm{e}^{At}\| -1)/t\le (\mathrm{e}^{\theta t}-1)/t for all t\ge0 and taking \lim_{t\to 0+} we conclude that \mu(A) \le (d/dt) \mathrm{e}^{\theta t}\mid_{t = 0} \,= \theta.

The logarithmic norm was introduced by Dahlquist (1958) and Lozinskii (1958) in order to obtain error bounds for numerical methods for solving differential equations. The bound (2) can alternatively be proved by using differential inequalities (see Söderlind (2006)). Not only is (2) sharper than \|\mathrm{e}^{At}\| \le \mathrm{e}^{\|A\|t}, but \mathrm{e}^{\|A\|t} is increasing in t while \mathrm{e}^{\mu(A)t} potentially decays, since \mu(A) < 0 is possible.

The spectral abscissa is defined by

\notag      \alpha(A) = \max \{\, \mathrm{Re} \lambda : \lambda \in \Lambda(A)\,\},

where \Lambda(A) denotes the spectrum of A (the set of eigenvalues). Whereas the norm bounds the spectral radius (\rho(A) \le \|A\|), the logarithmic norm bounds the spectral abscissa, as shown by the next result.

Theorem 4. For A\in\mathbb{C}^{n\times n} and a consistent matrix norm,

\notag    -\mu(-A) \le \alpha(A) \le \mu(A).

Combining (1) with (2) gives

\notag   -\|A\| \le -\mu(-A) \le \alpha(A) \le \mu(A) \le \|A\|.

In view of Lemma 1, the logarithmic norm \mu(A) can be expressed as the one-sided derivative of \|\mathrm{e}^{tA}\| at t = 0, so \mu(A) determines the initial rate of change of \|\mathrm{e}^{tA}\| (as well as providing the bound \mathrm{e}^{\mu(A)t} for all t). The long-term behavior is determined by the spectral abscissa \alpha(A), since \|\mathrm{e}^{tA}\| \to 0 as t\to\infty if and only if \alpha(A) < 0, which can be proved using the Jordan canonical form.

The next result provides a bound on the norm of the inverse of a matrix in terms of the logarithmic norm.

Theorem 5. For a nonsingular matrix A\in\mathbb{C}^{n\times n} and a subordinate matrix norm, if \mu(A) < 0 or \mu(-A) < 0 then

\notag    \|A^{-1}\| \le \displaystyle\frac{1}{\max\bigl(-\mu(A),-\mu(-A)\bigr)}.     \qquad (3)

Formulas for Logarithmic Norms

Explicit formulas are available for the logarithmic norm s corresponding to the 1, 2, and \infty-norms.

Theorem 6. For A\in\mathbb{C}^{n\times n},

\notag \begin{aligned}  \mu_1(A) &= \max_j \biggl( \sum_{i\ne j} |a_{ij}| +                                   \mathrm{Re}\, a_{jj} \biggr ),\\  \mu_{\infty}(A) &= \max_i \biggl( \sum_{j\ne i} |a_{ij}| +                                   \mathrm{Re}\, a_{ii} \biggr ),\\  \mu_2(A) &= \lambda_{\max}\Bigl( \frac{A+A^*}{2} \Bigr), \qquad (4) \end{aligned}

where \lambda_{\max} denotes the largest eigenvalue of a Hermitian matrix.

Proof. We have

\notag \begin{aligned}     \mu_{\infty}(A)     &= \lim_{\epsilon \to 0+} \frac{ \|I + \epsilon A\|_{\infty} - 1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \max_i \bigl(\sum_{j\ne i} |\epsilon a_{ij}|                               + |1 + \epsilon a_{ii}| \bigr) -1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \max_i \bigl(\sum_{j\ne i} |\epsilon a_{ij}|                   + \sqrt{(1 + \epsilon\mathrm{Re}\, a_{ii})^2                   + (\epsilon\mathrm{Im}\,a_{ii})^2}\, \bigr) -1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \max_i \bigl( \sum_{j\ne i} |\epsilon a_{ij}|                   + \epsilon\mathrm{Re}\, a_{ii} + O(\epsilon^2)  \bigr)}{\epsilon}\\     &= \max_i \biggl( \sum_{j\ne i} |a_{ij}| +                                   \mathrm{Re}\, a_{ii} \biggr ). \end{aligned}

The formula for \mu_1(A) follows, since \|A\|_1 = \|A^*\|_\infty implies \mu_1(A) = \mu_{\infty}(A^*). For the 2-norm, using \|A\|_2 = \rho(A^*A)^{1/2}, we have

\notag \begin{aligned}     \mu_2(A)     &= \lim_{\epsilon \to 0+} \frac{ \|I + \epsilon A\|_2 - 1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \rho\bigl( I + \epsilon(A+A^*) +       \epsilon^2 A^*A\bigr)^{1/2} -1} {\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ 1 + \frac{1}{2}\epsilon\lambda_{\max}(A+A^*) +                                 O(\epsilon^2)  -1} {\epsilon}\\     &= \lambda_{\max}\Bigl( \frac{A+A^*}{2} \Bigr). \end{aligned}

As a special case of (4), if A is normal, so that A = QDQ^* with Q unitary and D = \mathrm{diag}(\lambda_i), then \mu_2(A) = \max_i (\lambda_i + \overline{\lambda_i})/2 = \max_i \mathrm{Re} \lambda_i = \alpha(A).

We can make some observations about \mu_\infty(A) for A\in\mathbb{R}^{n\times n}.

  • If A\ge 0 then \mu_\infty(A) = \|A\|_\infty.
  • \mu_\infty(A) < 0 if and only if a_{ii} < 0 for all i and A is strictly row diagonally dominant.
  • For the \infty-norm the bound (3) is the same as a bound based on diagonal dominance except that it is applicable only when the diagonal is one-signed.

For a numerical example, consider the n\times n tridiagonal matrix anymatrix('gallery/lesp'), which has the form illustrated for n = 6 by

\notag     A_6 = \left[\begin{array}{cccccc}      -5 & 2 & 0 & 0 & 0 & 0\\      \frac{1}{2} & -7 & 3 & 0 & 0 & 0\\      0 & \frac{1}{3} & -9 & 4 & 0 & 0\\      0 & 0 & \frac{1}{4} & -11 & 5 & 0\\      0 & 0 & 0 & \frac{1}{5} & -13 & 6\\      0 & 0 & 0 & 0 & \frac{1}{6} & -15      \end{array}\right].

We find that \alpha(A_6) = -4.55 and \mu_2(A_6) = -4.24, and it is easy to see that \mu_1(A_n) = -4.5 and \mu_{\infty}(A_n) = -3 for all n. Therefore Theorem 4 shows that \mathrm{e}^{At}\to 0 as t\to \infty and \mu_1 gives a faster decaying bound than \mu_2 and \mu_\infty.

Now consider the subordinate matrix norm \|\cdot\|_G based on the vector norm \|x\|_G = (x^*Gx)^{1/2}, where G is a Hermitian positive definite matrix. The corresponding logarithmic norm \mu_G can be expressed as the largest eigenvalue of a Hermitian definite generalized eigenvalue problem.

Theorem 7. For A\in\mathbb{C}^{n\times n},

\notag    \mu_G(A) = \max\{\, \lambda: \det(GA + A^*G - 2\lambda G) = 0 \,\}.

Theorem 7 allows us to make a connection with the theory of ordinary differential equations (ODEs)

\notag   y' = f(t,y), \quad f: \mathbb{R} \times \mathbb{R}^n\to \mathbb{R}^n.   \qquad (5)

Let G\in\mathbb{R}^{n\times n} be symmetric positive definite and consider the inner product \langle x, y \rangle = x^*Gy and the corresponding norm defined by \|x\|_G^2 = \langle x, x \rangle = (x^*Gx)^{1/2}. It can be shown that for A\in\mathbb{R}^{n\times },

\notag   \mu_G(A) = \max_x \displaystyle\frac{\langle Ax, x\rangle}{\|x\|_G^2}.  \qquad (6)

The function f satisfies a one-sided Lipschitz condition if there is a function v(t) such that

\langle f(t,y) - f(t,z), y - z \rangle \le v(t) \|y-z\|^2

for all y,z in some region and all a\le t \le b. For the linear differential equation with f(t,y) = A(t)y in (5), using (6) we obtain

\langle f(t,y) - f(t,z), y - z \rangle =    \langle A(t)(y-z), y - z \rangle \le \mu_G(A(t)) \|y-z\|_G^2,

and so the logarithmic norm \mu_G(A(t)) can be taken as a one-sided Lipschitz constant. This observation leads to results on contractivity of ODEs; see Lambert (1991) for details.

Notes

For more on the use of the logarithmic norm in differential equations see Coppel (1965) and Söderlind (2006).

The proof of Theorem 3 is from Hinrichsen and Pritchard (2005).

References

This is a minimal set of references, which contain further useful references within.

  • W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations}. D. C. Heath and Company, Boston, MA. USA, 1965.
  • Germund Dahlquist. Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. PhD thesis, Royal Inst. of Technology, Stockholm, Sweden, 1958.
  • Diederich Hinrichsen and Anthony J. Pritchard. Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness. Springer-Verlag, Berlin, Germany, 2005.
  • J. D. Lambert. Numerical Methods for Ordinary Differential Systems. The Initial Value Problem. John Wiley, Chichester, 1991.
  • Gustaf Söderlind, The Logarithmic Norm. History and Modern Theory. BIT, 46(3):631–652, 2006.
  • Torsten Ström. On Logarithmic Norms. SIAM J. Numer. Anal., 12(5):741–753, 1975.

Related Blog Posts

This article is part of the “What Is” series, available from https://nhigham.com/index-of-what-is-articles/ and in PDF form from the GitHub repository https://github.com/higham/what-is.

What Is a Tridiagonal Matrix?

A tridiagonal matrix is a square matrix whose elements are zero away from the main diagonal, the subdiagonal, and the superdiagonal. In other words, it is a banded matrix with upper and lower bandwidths both equal to 1. It has the form

\notag    A =  \begin{bmatrix}          d_1 & e_1 &        &        &\\          c_2 & d_2 & e_2    &        &\\              & c_3 & \ddots & \ddots &\\              &     & \ddots & \ddots & e_{n-1}\\              &     &        & c_n    & d_n          \end{bmatrix} \in\mathbb{C}^{n\times n}.

An important example is the matrix T_n that arises in discretizating the Poisson partial differential equation by a standard five-point operator, illustrated for n = 5 by

\notag    T_5 = \left[    \begin{array}{@{}*{4}{r@{\mskip10mu}}r}                 4  & -1 &    &    &    \\                 -1 & 4  & -1 &    &    \\                    & -1 & 4  & -1 &    \\                    &    & -1 & 4  & -1 \\                    &    &    & -1 & 4    \end{array}\right].

It is symmetric positive definite, diagonally dominant, a Toeplitz matrix, and an M-matrix.

Tridiagonal matrices have many special properties and various algorithms exist that exploit their structure.

Symmetrization

It can be useful to symmetrize a matrix by transforming it with a diagonal matrix. The next result shows when symmetrization is possible by similarity.

Theorem 1. If A\in\mathbb{R}^{n\times n} is tridiagonal and c_i e_{i-1} > 0 for all i then there exists a diagonal D\in\mathbb{R}^{n\times n} with positive diagonal elements such that D^{-1}AD is symmetric, with (i-1,i) element (c_ie_{i-1})^{1/2}.

Proof. Let D = \mathrm{diag}(\omega_i). Equating (i,i-1) and (i-1,i) elements in the matrix D^{-1}AD gives

\notag  c_i \displaystyle\frac{\omega_{i-1}}{\omega_i} =  e_{i-1} \displaystyle\frac{\omega_i}{\omega_{i-1}}, \quad i = 2:n, \qquad(1)

or

\notag  \biggl(\displaystyle\frac{\omega_{i-1}}{\omega_i}\biggr)^2 =  \frac{e_{i-1} }{c_i}, \quad i = 2:n. \qquad (2)

As long as c_i e_{i-1} > 0 for all i we can set \omega_1 = 1 and solve (2) to obtain real, positive \omega_i, i = 2:n. The formula for the off-diagonal elements of the symmetrized matrix follows from (1) and (2).

LU Factorization

The LU factors of a tridiagonal matrix are bidiagonal:

\notag     L =   \begin{bmatrix}                     1    &        &        &        & \\                   \ell_2 & 1      &        &        & \\                          & \ell_3 & 1      &        & \\                          &        & \ddots & \ddots & \\                          &        &        & \ell_n & 1                       \end{bmatrix},     \quad    U = \begin{bmatrix}                 u_1 &  e_1 &        &        &         \\                      &  u_2 &  e_2   &        &         \\                      &      & \ddots & \ddots &         \\                      &      &        & \ddots &  e_{n-1}\\                      &      &        &        & u_n        \end{bmatrix}. \qquad (3)

The equation A = LU gives the recurrence

\notag  u_1 = d_1, \qquad  \ell_i = c_i/u_{i-1}, \quad                   u_i = d_i - \ell_i e_{i-1}, \quad                     i=2\colon n. \qquad (4)

The recurrence breaks down with division by zero if one of the leading principal submatrices A(1:k,1:k), k = 1:n-1, is singular. In general, partial pivoting must be used to ensure existence and numerical stability, giving a factorization PA = LU where L has at most two nonzeros per column and U has an extra superdiagonal. The growth factor \rho_n is easily seen to be bounded by 2.

For a tridiagonal Toeplitz matrix

\notag  T_n(c,d,e) = \begin{bmatrix}                      d   & e      &        &   \\                        c & d      & \ddots &   \\                          & \ddots & \ddots & e \\                          &        & c      & d                \end{bmatrix} \in\mathbb{C}^{n\times n} \qquad (5)

the elements of the LU factors converge as n\to\infty if A is diagonally dominant.

Theorem 2. Suppose that T_n(c,d,e) has an LU factorization with LU factors (3) and that ce > 0 and |d| \ge 2\sqrt{ce}. Then |u_j| decreases monotonically and

\notag    \lim_{n\to \infty}u_n   = \frac{1}{2} \Bigl(d + \mathrm{sign}(d)     \sqrt{ d^2 - 4ce } \Bigr).

From (4), it follows that under the conditions of Theorem 2, |\ell_i| increases monotonically and \lim_{n\to\infty}\ell_n = e/\lim_{n\to\infty}u_n. Note that the conditions of Theorem 2 are satisfied if A is diagonally dominant by rows, since ce> 0 implies d \ge c + e \ge 2\sqrt{ce}. Note also that if we symmetrize A using Theorem 1 then we obtain the matrix T_n(\sqrt{ce},d, \sqrt{ce}), which is irreducibly diagonally dominant and hence positive definite if d > 0.

Inverse

The inverse of a tridiagonal matrix is full, in general. For example,

\notag   T_5(-1,3,-1)^{-1}   = \left[\begin{array}{rrrrr} 3 & -1 & 0 & 0 & 0\\                               -1 & 3 & -1 & 0 & 0\\                               0 & -1 & 3 & -1 & 0\\                               0 & 0 & -1 & 3 & -1\\                               0 & 0 & 0 & -1 & 3 \end{array}\right]^{-1}  =  \frac{1}{144}     \left[\begin{array}{ccccc} 55 & 21 & 8  & 3  & 1 \\                                21 & 63 & 24 & 9  & 3 \\                                8 & 24 & 64 & 24 & 8 \\ 3 & 9 & 24 & 63 & 21\\                                1 & 3  & 8  & 21 & 55 \end{array}\right].

Since an n\times n tridiagonal matrix depends on only 3n-2 parameters, the same must be true of its inverse, meaning that there must be relations between the elements of the inverse. Indeed, in T_5(-1,3,-1)^{-1} any 2\times 2 submatrix whose elements lie in the upper triangle is singular, and the (1:3,3:5) submatrix is also singular. The next result explains this special structure. We note that a tridiagonal matrix is irreducible if a_{i+1,i} and a_{i,i+1} are nonzero for all i.

Theorem 3. If A\in\mathbb{C}^{n\times n} is tridiagonal, nonsingular, and irreducible then there are vectors u, v, x, and y, all of whose elements are nonzero, such that

\notag           (A^{-1})_{ij} = \begin{cases}                                 u_iv_j, & i\le j,\\                                 x_iy_j, & i\ge j.\\[4pt]                           \end{cases}

The theorem says that the upper triangle of the inverse agrees with the upper triangle of a rank-1 matrix (uv^T) and the lower triangle of the inverse agrees with the lower triangle of another rank-1 matrix (xy^T). This explains the singular submatrices that we see in the example above.

If a tridiagonal matrix A is reducible with a_{k,k+1} = 0 then it has the block form

\notag     \begin{bmatrix}       A_{11}   & 0 \\       A_{21}   & A_{22}     \end{bmatrix},

where A_{21} = a_{k+1,k}e_1e_k^T, and so

\notag     \begin{bmatrix}       A_{11}^{-1}   & 0 \\       - A_{22}^{-1} A_{21} A_{11}^{-1}  & A_{22}^{-1}     \end{bmatrix},

in which the (2,1) block is rank 1 if a_{k+1,k}\ne 0. This blocking can be applied recursively until Theorem 1 can be applied to all the diagonal blocks.

The inverse of the Toeplitz tridiagonal matrix T_n(a,b,c) is known explicitly; see Dow (2003, Sec. 3.1).

Eigenvalues

The most widely used methods for finding eigenvalues and eigenvectors of Hermitian matrices reduce the matrix to tridiagonal form by a finite sequence of unitary similarity transformations and then solve the tridiagonal eigenvalue problem. Tridiagonal eigenvalue problems also arise directly, for example in connection with orthogonal polynomials and special functions.

The eigenvalues of the Toeplitz tridiagonal matrix T_n(c,d,e) in (5) are given by

\notag         d + 2 (ce)^{1/2} \cos\biggl( \displaystyle\frac{k \pi}{n+1} \biggr),         \quad k = 1:n. \qquad (6)

The eigenvalues are also known for certain variations of the symmetric matrix T_n(c,d,c) in which the (1,1) and (n,n) elements are modified (Gregory and Karney, 1969).

Some special results hold for the eigenvalues of general tridiagonal matrices. A matrix is derogatory if an eigenvalue appears in more than one Jordan block in the Jordan canonical form, and nonderogatory otherwise.

Theorem 4. If A\in\mathbb{C}^{n\times n} is an irreducible tridiagonal matrix then it is nonderogatory.

Proof. Let G = A - \lambda I, for any \lambda. The matrix G(1:n-1,2:n) is lower triangular with nonzero diagonal elements e_1, \dots, e_{n-1} and hence it is nonsingular. Therefore G has rank at least n-1 for all \lambda. If A were derogatory then the rank of G would be at most n-2 when \lambda is an eigenvalue, so A must be nonderogatory.

Corollary 5. If A\in\mathbb{R}^{n\times n} is tridiagonal with c_i e_{i-1} > 0 for all i then the eigenvalues of A are real and simple.

Proof. By Theorem 1 the eigenvalues of A are those of the symmetric matrix S = D^{-1}AD and so are real. The matrix S is tridiagonal and irreducible so it is nonderogatory by Theorem 4, which means that its eigenvalues are simple because it is symmetric.

The formula (6) confirms the conclusion of Corollary 5 for tridiagonal Toeplitz matrices.

Corollary 5 guarantees that the eigenvalues are distinct but not that they are well separated. The spacing of the eigenvalues in (6) clearly reduces as n increases. Wilkinson constructed a symmetric tridiagonal matrix called W_n^+, defined by

\notag  \begin{aligned}    d_i &= \frac{n+1}{2} - i = d_{n-i+1}, \quad i = 1:n/2, \qquad    d_{(n+1)/2} = 0 \quad \mathrm{if}~n~\mathrm{is~odd}, \\    c_i &= e_{i-1} = 1.  \end{aligned}

For example,

\notag   W_5^+ = \left[\begin{array}{ccccc} 2 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 1 & 2 \end{array}\right].

Here are the two largest eigenvalues of W_{21}^+, as computed by MATLAB.

>> A = anymatrix('matlab/wilkinson',21); 
>> e = eig(A); e([20 21])
ans =
  10.746194182903322
  10.746194182903393

These eigenvalues (which are correct to the digits shown) agree almost to the machine precision.

Notes

Theorem 2 is obtained for symmetric matrices by Malcolm and Palmer (1974), who suggest saving work in computing the LU factorization by setting u_j = u_k for j > k once u_k is close enough to the limit.

A sample reference for Theorem 3 is Ikebe (1979), which is one of many papers on inverses of banded matrices.

The eigenvectors of a symmetric tridiagonal matrix satisfy some intricate relations; see Parlett (1998, sec. 7.9).

References

This is a minimal set of references, which contain further useful references within.